# Plotting points in three dimensions

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25:20:24 of on-demand video • Updated November 2020

- Partial Derivatives, including higher order partial derivatives, multivariable chain rule and implicit differentiation
- Multiple Integrals, including approximating double and triple integrals, finding volume, and changing the order of integration
- Vectors, including derivatives and integrals of vector functions, arc length and curvature, and line and surface integrals

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Today we're going to be talking about plotting points in three dimensional coordinate space. And in this particular problem we've been given for coordinate points each of which are defined in three dimensional coordinate space and we've been asked to plot them in three dimensional coordinate space. I've gone ahead and drawn four different sets of three dimensional axes all of which are going to follow the right hand rule where we have our axes or coordinate axes set up as x y and z this way and I'm going to plot one of these points on each set of axes. Now before we keep going because this is the very beginning of three dimensional coordinate space moving from two dimensional coordinate space. I just want to make a note that for anybody who might be confused remember that in two dimensional coordinate space we just had two variables x and y and we would have a two dimensional coordinate system like this with x and y. And if we had a point let's say we called or point to three then we would move out a distance of two along the x axis and then up a distance of three along the y axis and our coordinate point here would be at this point right here two three like this. Well this is going to be the same concept for three dimensional coordinate space except that we have to deal with a third dimension or third variable which is z and z is really just adding a third dimension a height component to this flat x y coordinate plane. So if we want to take this point to three and we wanted to plot it in three dimensional coordinate space what we would need to assume is that we were actually moving to the point to 3 0 where 0 is our z coordinate. We're not going to be moving up a distance along the z axis at all. We're going to be staying in the x y coordinate plane for what we would do is we'd come out a distance of two along the x axis here and then we would move parallel to the y axis up a distance of three. Let's say right here and that point to three would be here in our three dimensional coordinate system with no height component here in the z axis. We wouldn't move up along the z axis or down among z axis we would stay in the x y coordinate plane. What do you mean by coordinate planes is we have three coordinate planes here we have three coordinate axes. The x axis the y axis and the z axis. We then have three coordinate planes. One is the x y coordinate plane and the x y coordinate plane is this plane. Here this flat plane that includes the x axis and the y axis but not the z axis. Any point that lies in the x y coordinate plane will have a value as the value in the coordinates point of 0 and this x y coordinate plane is the same two dimensional x y coordinate plane that we're used to seeing when we have two variables. That's why we could take this point to three add a z value z coordinate point here of 0 and keep this point in the x y coordinate plane. This is our x y coordinate plane. We also have the y z coordinate plane which is this plane. Here if I draw lines the dotted lines dash lines that are parallel to each of these axes. This plane includes the z axis and the y axis. But every point that lies in this plane here will have an x value in its coordinate point of zero. This is the y z coordinate plane. And our third coordinate plane here is the x z coordinate plane right here. And every point that lies in this coordinate plane will have a Y value of zero. This is the x z coordinate plane. And of course we can have points that don't lie exactly in each of these three planes but lie somewhere in between them. If any value if any value in in this coordinate point is zero. I know it's going to lie in one of our coordinate planes So for example this first point 0 5 2 because the x value is zero. I know that it's going to lie in the y z plane. This point here because the y value is zero. I know it's going to lie in the x z plane. These two points two 4:6 and one negative 1 2 are not going to lie in any one of our three coordinate planes because none of their component values are equal to zero to four and six are not equal to zero and one negative one and two are all not equal to zero. So these two points are not going to lie in coordinate planes. So let's talk about now plotting points in this three dimensional coordinate system. Well let me first erase this little point that we drew as a translation from our x y coordinate plane. If I look at the point 0 5 to what I know is that I'm moving out a distance along the x axis of 0. Right. These points are all x y z. I'm moving out a distance along the x axis of zero because the x value is zero. So what that tells me is that I started the origin. I'm not going to move any distance along the x axis because my x value is zero. So from the origin still I'm going to move our distance along the y axis of 5 so I'm going to go out here to let's say 5. Can I go out here to five what's called that y equals five. And then my z value is positive too which means I'm going to move up a distance along the z axis or parallel to the z axis of 2. Let's say that's a value of 2 there. And I'm going to plot my point here right here. This is the point 0 5 2. It lies in the y z plane so I'm going to say 0 5 2. It lies in the y z plane the x value is 0 and what I like to do sometimes just to clarify is I like to draw corresponding parallel lines to make a rectangle out of this because this lies in one of our coordinate planes. We're just going to see a rectangle here but our last two points that don't line coordinate planes we're going to see a box. But here we see a rectangle that gives us a better picture of exactly where this point is sometimes it can be hard to visualize in three dimensional space but drawing a rectangle like this that goes out to this point helps us visualize exactly where it is. Now let's look at for zero negative 1. Well in this case I'm going to come out a distance along the x axis of for let's say that that's a distance of 4 along the x axis the y value is zero. So if I have a non-zero y value I would move out parallel to the y axis this way or if the Y that was negative out parallel to the y axis this way but because it's zero I'm going to stick here along my x axis. Then I move a distance of negative 1 along the z axis on the z value is positive. I move up parallel to the z axis. When it's negative I move down parallel to the z axis a distance of 1. And there's my point right there. I can go ahead and label it as for zero negative 1. And because it's going to lie in the x z coordinate plane I can just draw a rectangle here that illustrates that x z coordinate plane and I know that my point is in that plane to 4:6 get a little more interesting. I move out a distance of two along the x axis like this. Then I move out a distance of four parallel to the y axis. Well the positive direction of the y axis is this way so I'm going to move out a distance of four should be double the distance of the X distance here because I have two and then four. So let's call that four and then I'm going to move up a distance of six parallel to the z axis. This is the positive direction of the Z axis this way. So I'm going to move up if I move up a distance of six. Let's say that that's six right there. Then here's my point. Two for six. Now if I really wanted to illustrate where that point is one thing that I could do is just go ahead and draw a rectangle that coincides with that point and the origin this point is that one vertex of our rectangular box here and another vertex of the box is at the origin so I can go ahead and draw this box right here. There's this other point at the origin right here. But basically my box looks like this if I draw dash lines back here. But there is that box and see how drawing the box right there makes this point pop out and I can really see where it is. I can label it as two four six but if I didn't draw that box if I took those green lines away it kind of be floating I wouldn't really be able to tell exactly where it was. Here I can tell it's in the first octant of this three dimensional coordinate system here. And same idea here when we're graphing the point one negative one too. We have our x value which is positive. So we want to move out on the positive direction of the x axis which is out this way. So we're going to move out a distance of one let's say that that's one right there then our Y value is negative 1. So this over here is the positive direction of the y axis. So since our Y value is negative we're going to move parallel to the y axis but away from the positive direction we're going to move in the negative direction out a distance of 1 so to say that that's there. And then our z value is positive. This is the positive direction of our z axis. So we're going to move up two units parallel to the z axis are going to move parallel to the z axis. Up to units of things this to right there. And here's our coordinate point one negative one too. Now we want to fill this out and draw our box so we get a better perspective of exactly where this point is. We can fill in the other sides of our box like this along the z axis and like this and like this and we can draw dotted lines here to represent the back of our box like this and that gives us a much clearer picture of exactly where our coordinate point is in relation to our coordinate axes and we can label it as one negative one too. You should always label the co-ordinate point after you sketch it like that. So that's just a basic overview of how to plot points in three dimensional coordinate space.